Problem: The $4$ points plotted below are on the graph of $y=b^x$. Based only on these $4$ points, plot the $4$ corresponding points that must be on the graph of $y=\log_b{x}$ by clicking on the graph. Click to add points
Solution: Let's consider the point on $y = { b}^ x$ with coordinates $( 1,D 3)$. Since $ y = \log_{ b}{ x}$ is the inverse of $ y={ b}^ x$, the point $(D 3, 1 )$ is on the graph of $ y = \log_{ b}{ x}$. In general, if $( p,D q)$ is on $ y={b}^ x$, then $( q,p )$ is on $ y = \log_{ b}{ x}$. For each point on $y=b^x$, we just switch the order of its coordinates to get a point on $y=\log_b{x}$. So, $y=\log_b{x}$ also has points with coordinates $(1, 0)$, $(9,2)$, and $(27, 3)$. Given the points that we know are on ${y=b^x}$, the graph below shows the $4$ points that must be on ${y=\log_b{x}}$. The original $4$ points are also plotted for reference. ${4}$ ${8}$ ${12}$ ${16}$ ${20}$ ${24}$ ${28}$ ${4}$ ${8}$ ${12}$ ${16}$ ${20}$ ${24}$ ${28}$ $y$ $x$